QIC Abstracts

 Vol.16 No.5&6, April 1, 2016

Research Articles:

Normalizer circuits and a Gottesman-Knill theorem for infinite-dimensional systems (pp0361-0422)
Juan Bermejo-Vega, Cedric Yen-Yu Lin, Maarten Van den Nest
Normalizer circuits are generalized Clifford circuits that act on arbitrary finite-dimensional systems $\mathcal{H}_{d_1}\otimes \cdots \otimes \mathcal{H}_{d_n}$ with a standard basis labeled by the elements of a finite Abelian group $G=\mathbb{Z}_{d_1}\times\cdots \times \mathbb{Z}_{d_n}$. Normalizer gates implement operations associated with the group $G$ and can be of three types: quantum Fourier transforms, group automorphism gates and quadratic phase gates. In this work, we extend the normalizer formalism \cite{VDNest_12_QFTs,BermejoVega_12_GKTheorem} to \emph{infinite dimensions}, by allowing normalizer gates to act on systems of the form $\mathcal{H}_\mathbb{Z}^{\otimes a}$: each factor $\mathcal{H}_\mathbb{Z}$ has a standard basis labeled by \emph{integers} $\mathbb{Z}$, and a Fourier basis labeled by \emph{angles}, elements of the \emph{circle group} $\mathbb{T}$. Normalizer circuits become hybrid quantum circuits acting both on continuous- and discrete-variable systems. We show that infinite-dimensional normalizer circuits can be efficiently simulated classically with a generalized \textbf{\emph{stabilizer formalism}} for Hilbert spaces associated with groups of the form $\mathbb{Z}^a\times \mathbb{T}^b \times \mathbb{Z}_{d_1}\times\cdots\times \mathbb{Z}_{d_n}$. We develop new techniques to track stabilizer-groups based on \emph{normal forms} for group automorphisms and quadratic functions. We use our normal forms to reduce the problem of simulating normalizer circuits to that of finding general solutions of systems of mixed real-integer linear equations \cite{BowmanBurget74_systems-Mixed-Integer_Linear_equations} and exploit this fact to devise a robust simulation algorithm: the latter remains efficient even in pathological cases where stabilizer groups become \emph{infinite}, \emph{uncountable} and \emph{non-compact}. The techniques developed in this paper might find applications in the study of fault-tolerant quantum computation with superconducting qubits.

Constructions of $q$-ary entanglement-assisted quantum MDS codes with minimum distance greater than q+1 (pp0423-0434)
Jihao Fan, Hanwu Chen, and Juan Xu
The entanglement-assisted stabilizer formalism provides a useful framework for constructing quantum error-correcting codes (QECC), which can transform arbitrary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using pre-shared entanglement between the sender and the receiver. In this paper, we construct five classes of entanglement-assisted quantum MDS (EAQMDS) codes based on classical MDS codes by exploiting one or more pre-shared maximally entangled states. We show that these EAQMDS codes have much larger minimum distance than the standard quantum MDS (QMDS) codes of the same length, and three classes of these EAQMDS codes consume only one pair of maximally entangled states.

Multiparty quantum signature schemes (pp0435-0464)
Juan Miguel Arrazola, Petros Wallden, and Erika Andersson
Digital signatures are widely used in electronic communications to secure important tasks such as financial transactions, software updates, and legal contracts. The signature schemes that are in use today are based on public-key cryptography and derive their security from computational assumptions. However, it is possible to construct unconditionally secure signature protocols. In particular, using quantum communication, it is possible to construct signature schemes with %information-theoretic security based on fundamental principles of quantum mechanics. Several quantum signature protocols have been proposed, but none of them has been explicitly generalised to more than three participants, and their security goals have not been formally defined. Here, we first extend the security definitions of Swanson and Stinson \cite{swanson2011unconditionally} so that they can apply also to the quantum case, and introduce a formal definition of transferability based on different verification levels. We then prove several properties that multiparty signature protocols with information-theoretic security -- quantum or classical -- must satisfy in order to achieve their security goals. We also express two existing quantum signature protocols with three parties in the security framework we have introduced. Finally, we generalize a quantum signature protocol given in \cite{dunjko2014QDSQKD} to the multiparty case, proving its security against forging, repudiation and non-transferability. Notably, this protocol can be implemented using any point-to-point quantum key distribution network and therefore is ready to be experimentally demonstrated.

Realizing an N-two-qubit quantum logic gate in a cavity QED with nearest qubit--qubit interaction (pp0465-0482)
Taoufik Said, Abdelhaq Chouikh, Karima Essammouni, and Mohamed Bennai
We propose an effective way for realizing a three quantum logic gates (NTCP gate, NTCP-NOT gate and NTQ-NOT gate) of one qubit simultaneously controlling N target qubits based on the qubit-qubit interaction. We use the superconducting qubits in a cavity QED driven by a strong microwave field. In our scheme, the operation time of these gates is independent of the number N of qubits involved in the gate operation. These gates are insensitive to the initial state of the cavity QED and can be used to produce an analogous CNOT gate simultaneously acting on N qubits. The quantum phase gate can be realized in a time (nanosecond-scale) much smaller than decoherence time and dephasing time (microsecond-scale) in cavity QED. Numerical simulation under the influence of the gate operations shows that the scheme could be achieved efficiently within current state-of-the-art technology.

Non-Markovian quantum trajectroy unravellings of entanglement (pp0483-0497)
Brittany Corn, Jun Jing, and Ting Yu
The fully quantized model of double qubits coupled to a common bath is solved using the quantum state diffusion (QSD) approach in the non-Markovian regime. We have established the explicit time-local non-Markovian QSD equations for the two-qubit dissipative and dephasing models. Diffusive quantum trajectories are applied to the entanglement estimation of two-qubit systems in a non-Markovian regime. In both cases, non-Markovian features of entanglement evolution are revealed through quantum diffusive unravellings in the system state space.

Quantum interpretations of AWPP and APP (pp0498-0514)
Tomoyuki Morimae and Harumichi Nishimura
AWPP is a complexity class introduced by Fenner, Fortnow, Kurtz, and Li, which is defined using GapP functions. Although it is an important class as the best upperbound of BQP, its definition seems to be somehow artificial, and therefore it would be better if we have some ``physical interpretation" of AWPP. Here we provide a quantum physical interpretation of AWPP: we show that AWPP is equal to the class of problems efficiently solved by a quantum computer with the ability of postselecting an event whose probability is close to an FP function. This result is applied to also obtain a quantum physical interpretation of APP. In addition, we consider a ``classical physical analogue" of these results, and show that a restricted version of ${\rm BPP}_{\rm path}$ contains ${\rm UP}\cap{\rm coUP}$ and is contained in WAPP.

A localized quantum walk with a gap in distribution (pp0515-0529)
Takuya Machida
Quantum walks behave differently from what we expect and their probability distributions have unique structures. They have localization, singularities, a gap, and so on. Those features have been discovered from the view point of mathematics and reported as limit theorems. In this paper we focus on a time-dependent three-state quantum walk on the line and demonstrate a limit distribution. Three coin states at each position are iteratively updated by a coin-flip operator and a position-shift operator. As the result of the evolution, we end up to observe both localization and a gap in the limit distribution.

Entanglement of simultaneous and non-simultaneous accelerated qubit-qutrit systems (pp0530-0542)
Nasser Metwally
 In this contribution, we investigate the entanglement behavior of a composite system consists of two different dimensional subsystems in non-inertial frames. In particular, we consider a composite system of a qubit(two-dimensional) subsystem and a qutrit (three-dimensional) subsystem. The degree of entanglement is quantified for different cases, where it is assumed that the two-subsystems are simultaneously or non-simultaneously accelerated. The entanglement decays with the increase of the acceleration of any subsystem. In general, the decay rate of entanglement increases with higher dimension of the accelerated subsystem. These results could be important in building an accelerated quantum network consist of different dimensions nodes.

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